Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x1))) → A(a(a(x1)))
A(b(c(x1))) → A(c(c(a(a(a(x1))))))
A(b(c(x1))) → A(x1)
A(b(c(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x1))) → A(a(a(x1)))
A(b(c(x1))) → A(c(c(a(a(a(x1))))))
A(b(c(x1))) → A(x1)
A(b(c(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x1))) → A(a(a(x1)))
A(b(c(x1))) → A(x1)
A(b(c(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(c(x1))) → A(a(x1)) at position [0] we obtained the following new rules:
A(b(c(a(x0)))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(c(c(a(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x1))) → A(a(a(x1)))
A(b(c(b(c(x0))))) → A(a(c(c(a(a(a(x0)))))))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x1))) → A(a(a(x1)))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(c(x1))) → A(a(a(x1))) at position [0] we obtained the following new rules:
A(b(c(x0))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(x0)))) → A(a(b(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x0))) → A(b(x0))
A(b(c(a(x0)))) → A(a(b(x0)))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(c(a(x0)))) → A(a(b(x0))) at position [0] we obtained the following new rules:
A(b(c(a(c(x0))))) → A(a(c(c(a(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x0))) → A(b(x0))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(c(x0))))) → A(a(c(c(a(a(a(x0)))))))
A(b(c(x1))) → A(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
Q DP problem:
The TRS P consists of the following rules:
A(b(c(x0))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The finiteness of this DP problem is implied by strong termination of a SRS due to [12].
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
A(b(c(x0))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
a(b(c(x1))) → a(c(c(a(a(a(x1))))))
A(b(c(x0))) → A(b(x0))
A(b(c(b(c(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(b(c(a(x0)))) → A(b(x0))
A(b(c(x1))) → A(x1)
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
C(b(c(b(A(x))))) → A1(a(a(c(c(a(a(A(x))))))))
C(b(a(x))) → C(a(x))
C(b(a(x))) → A1(a(a(c(c(a(x))))))
C(b(c(b(A(x))))) → A1(a(A(x)))
C(b(c(b(A(x))))) → A1(a(c(c(a(a(A(x)))))))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(a(x))) → A1(a(c(c(a(x)))))
C(b(a(x))) → A1(c(c(a(x))))
C(b(c(b(A(x))))) → C(c(a(a(A(x)))))
C(b(a(x))) → C(c(a(x)))
C(b(c(b(A(x))))) → A1(A(x))
C(b(c(b(A(x))))) → A1(c(c(a(a(A(x))))))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(c(b(A(x))))) → A1(a(a(c(c(a(a(A(x))))))))
C(b(a(x))) → C(a(x))
C(b(a(x))) → A1(a(a(c(c(a(x))))))
C(b(c(b(A(x))))) → A1(a(A(x)))
C(b(c(b(A(x))))) → A1(a(c(c(a(a(A(x)))))))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(a(x))) → A1(a(c(c(a(x)))))
C(b(a(x))) → A1(c(c(a(x))))
C(b(c(b(A(x))))) → C(c(a(a(A(x)))))
C(b(a(x))) → C(c(a(x)))
C(b(c(b(A(x))))) → A1(A(x))
C(b(c(b(A(x))))) → A1(c(c(a(a(A(x))))))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(x))) → C(a(x))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(c(b(A(x))))) → C(c(a(a(A(x)))))
C(b(a(x))) → C(c(a(x)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(a(x))) → C(a(x)) at position [0] we obtained the following new rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(b(A(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(c(b(A(x))))) → C(c(a(a(A(x)))))
C(b(a(x))) → C(c(a(x)))
C(b(a(c(b(A(x0)))))) → C(b(A(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(c(b(A(x))))) → C(c(a(a(A(x)))))
C(b(a(x))) → C(c(a(x)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(c(b(A(x))))) → C(c(a(a(A(x))))) at position [0] we obtained the following new rules:
C(b(c(b(A(y0))))) → C(c(b(A(y0))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(c(b(A(y0))))) → C(c(b(A(y0))))
C(b(a(x))) → C(c(a(x)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(a(x))) → C(c(a(x))) at position [0] we obtained the following new rules:
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(a(a(x0)))) → C(c(b(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(c(b(A(x))))) → C(a(a(A(x))))
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(c(b(A(y0))))) → C(c(b(A(y0))))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(c(b(A(x))))) → C(a(a(A(x)))) at position [0] we obtained the following new rules:
C(b(c(b(A(y0))))) → C(b(A(y0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(c(b(A(y0))))) → C(c(b(A(y0))))
C(b(c(b(A(y0))))) → C(b(A(y0)))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(c(b(A(y0))))) → C(c(b(A(y0))))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(c(b(A(y0))))) → C(c(b(A(y0)))) at position [0] we obtained the following new rules:
C(b(c(b(A(x0))))) → C(A(x0))
C(b(c(b(A(x0))))) → C(b(A(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(c(b(A(x0))))) → C(A(x0))
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(c(b(A(x0))))) → C(b(A(x0)))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(c(b(A(x0))))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(a(c(b(A(x0)))))) → C(c(b(A(x0)))) at position [0] we obtained the following new rules:
C(b(a(c(b(A(x0)))))) → C(A(x0))
C(b(a(c(b(A(x0)))))) → C(b(A(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(c(b(A(x0)))))) → C(A(x0))
C(b(a(c(b(A(x0)))))) → C(b(A(x0)))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(a(x0)))) → C(b(x0))
C(b(a(a(x0)))) → C(c(b(x0)))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
a(b(c(x))) → a(c(c(a(a(a(x))))))
A(b(c(x))) → A(b(x))
A(b(c(b(c(x))))) → A(a(a(c(c(a(a(a(x))))))))
A(b(c(a(x)))) → A(b(x))
A(b(c(x))) → A(x)
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
a(b(c(x))) → a(c(c(a(a(a(x))))))
A(b(c(x))) → A(b(x))
A(b(c(b(c(x))))) → A(a(a(c(c(a(a(a(x))))))))
A(b(c(a(x)))) → A(b(x))
A(b(c(x))) → A(x)
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x)) → b(x)
c(b(a(x))) → a(a(a(c(c(a(x))))))
c(b(A(x))) → b(A(x))
c(b(c(b(A(x))))) → a(a(a(c(c(a(a(A(x))))))))
a(c(b(A(x)))) → b(A(x))
c(b(A(x))) → A(x)
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
a(b(c(x))) → a(c(c(a(a(a(x))))))
A(b(c(x))) → A(b(x))
A(b(c(b(c(x))))) → A(a(a(c(c(a(a(a(x))))))))
A(b(c(a(x)))) → A(b(x))
A(b(c(x))) → A(x)
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
a(b(c(x))) → a(c(c(a(a(a(x))))))
A(b(c(x))) → A(b(x))
A(b(c(b(c(x))))) → A(a(a(c(c(a(a(a(x))))))))
A(b(c(a(x)))) → A(b(x))
A(b(c(x))) → A(x)
Q is empty.